Figurate Numbers Calculator (Triangular, Square, Pentagonal, Hexagonal)
Calculate triangular, square, pentagonal, and hexagonal numbers from the 1st to the Nth term and compare them on a graph. Instantly compute the value of any nth term. Includes a reference table for terms 1-15.
Reference table for terms 1-15 (triangular, square, pentagonal, hexagonal)
A table listing the values of the four types of figurate numbers from the 1st to the 15th term.
| n | Triangular numbers | Square numbers | Pentagonal numbers | Hexagonal numbers |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 3 | 4 | 5 | 6 |
| 3 | 6 | 9 | 12 | 15 |
| 4 | 10 | 16 | 22 | 28 |
| 5 | 15 | 25 | 35 | 45 |
| 6 | 21 | 36 | 51 | 66 |
| 7 | 28 | 49 | 70 | 91 |
| 8 | 36 | 64 | 92 | 120 |
| 9 | 45 | 81 | 117 | 153 |
| 10 | 55 | 100 | 145 | 190 |
| 11 | 66 | 121 | 176 | 231 |
| 12 | 78 | 144 | 210 | 276 |
| 13 | 91 | 169 | 247 | 325 |
| 14 | 105 | 196 | 287 | 378 |
| 15 | 120 | 225 | 330 | 435 |
Tips
- Triangular numbers T(n) = n(n+1)/2 count the total number of points when stacked into an equilateral triangle. The classic bowling pin arrangement (10 pins) is triangular number T(4) = 10.
- Square numbers S(n) = n² count the total number of points arranged in a square. They are also known for the property that summing consecutive odd numbers produces a square number (1, 1+3=4, 1+3+5=9, ...).
- Pentagonal numbers P(n) = n(3n-1)/2 and hexagonal numbers H(n) = n(2n-1) count points arranged into a regular pentagon and hexagon respectively. Every hexagonal number is also a triangular number (H(n) = T(2n-1)).
- The graph lets you compare how fast each of the four types grows. For the same n, shapes with more vertices (3, 4, 5, 6) tend to produce larger values.
- This tool supports calculations up to the 1000th term. Since values grow rapidly for larger n, the graph and table display are limited to 1-30 terms for readability.
Frequently Asked Questions
Side Note — The Pythagoreans and the "mystery of numbers" found in figurate numbers
The study of figurate (polygonal) numbers dates back to the Pythagorean school of ancient Greece around the 6th century BCE. Guided by the belief that "all is number," they arranged pebbles into geometric shapes to visually understand the properties of numbers. The very names "triangular" and "square" numbers come from the shapes formed when the points are laid out.
There is a beautiful relationship between triangular and square numbers: adding any two consecutive triangular numbers always produces a square number (e.g., T(3)+T(4) = 6+10 = 16 = 4²). This becomes intuitively clear when you actually draw the arrangement of points. Geometric proof techniques like this had a major influence on the later development of number theory.
Figurate numbers remain an active subject of study in modern mathematics — for instance, the property that "every hexagonal number is also triangular," or the search for numbers that satisfy multiple figurate-number definitions at once, are still popular exercises in number theory today. It is remarkable that many properties the Pythagoreans discovered 2,500 years ago, arranging pebbles by hand with no paper or written arithmetic, still hold as valid theorems today.